Ray-Singer Torsion, Topological field theories and the Riemann zeta function at s=3

TL;DR

This paper explores the Ray-Singer analytic torsion in three-dimensional topological field theories, providing explicit formulas for lens spaces, analyzing their determinants, and deriving new results related to the Riemann zeta function at s=3.

Contribution

It introduces explicit analytic continuations of zeta functions for lens spaces and presents novel formulas for determinants and torsion, including new insights into zeta(3).

Findings

Torsion is trivial for L(6,1), L(10,3), and L(12,5).

Determinants and torsion grow with p for large p.

Multiple formulas for zeta(3) are derived.

Abstract

Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large $p$ behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for zeta(3): the ordinary Riemann zeta function evaluated at s=3. The torsion turns out to be trivial for the cases L(6,1), L((10,3) and L(12,5) and is, in general, greater than unity for large p and less than unity for a finite number of p and q.